This paper uses previous results of Chillingworth, Marsden and Wan on symmetry and bifurcation for the traction problem in three dimensional elastostatics to establish new results on the Signorini expansion. We show that the Signorini compatibility conditions are necessary and sufficient for linearization stability and analogies with results known for other field theories are pointed out. Under an explicit non-degeneracy condition, a new series expansion is given in which successive terms are inductively determined in pairs rather than singly. Our results include as special cases, classical results of Signorini, Tolotti and Stoppelli.

A notable achievement in the algebraic theory of semigroups has been the discovery by Nambooripad of the natural order on a regular semigroup. He has shown that this order is compatible with multiplication if and only if the semigroup is locally inverse, in the sense that every local submonoid is an inverse semigroup. In this paper we determine precisely when the natural order is compatible on the right (respectively left) with multiplication; this is so if and only if every local submonoid is ℒ-unipotent (respectively ℛ-unipotent).

We consider a common abstraction of de Morgan algebras and Stone algebras which we call an MS-algebra. The variety of MS-algebras is easily described by adjoining only three simple equations to the axioms for a bounded distributive lattice. We first investigate the elementary properties of these algebras, then we characterise the least congruence which collapses all the elements of an ideal, and those ideals which are congruence kernels. We introduce a congruence which is similar to the Glivenko congruence in a p-algebra and show that the location of this congruence in the lattice of congruences is closely related to the subdirect irreducibility of the algebra. Finally, we give a complete description of the subdirectly irreducible MS-algebras.

This is a study of the family of power series where Σ αnZn has unit radius of convergence and the εn are independent random variables taking the values ±1 with equal probability. It is shown that if

then almost all these power series take every complex value infinitely often in the unit disk.

In an earlier paper (1981), the present authors made a conjecture about the number of solutions of a semilinear elliptic boundary value problem which has been investigated extensively in the past decade. The conjecture is proved in the one-dimensional case.

This paper treats the global qualitative behaviour of all bifurcating configurations of whirling nonlinearly elastic strings with ends fixed on the axis of rotation.

A class of evolution problems is investigated [see (1.5), (1.6), (1.7) of the present paper] which includes, as a particular case, an evolution problem previously considered by a different author. Existence and uniqueness theorems are given in several function spaces. It is shown that, when the solution is required to belong to spaces of smooth functions, the problem becomes overdetermined. The necessary and sufficient integro-differential equations, to be satisfied by the datum, for the existence of a smooth solution, are given.

In §§1 and 2, we consider mainly a system of reaction-diffusion equations with general diffusion matrix and we establish the stabilization of all solutions at t →∞. The interest of this problem derives from two separate facts. First, the sets that are useful for localizing the asymptotics cease to be invariant as soon as the diffusion matrix is not a multiple of the identity. Second, the set of equilibria is connected. In §3, we establish uniform L§ bounds for the solutions of a class of parabolic systems. The unifying feature in the problems considered is the lack of any conventional maximum principles.

Let

be a cusp form of even integral weight k > 2 for the full modular group. Then the Dirichlet series

is absolutely convergent for σ > ½(k + 1). Hecke showed that LF is an entire function of s satisfying the functional equation

Barnes and Sloane recently described a “general construction” for lattice packings of equal spheres in Euclidean space. In the present paper we simplify and further generalize their construction, and make it suitable for iteration. As a result we obtain lattice packings in ℝm with density Δ satisfying , as m → ∞ where is the smallest value of k for which the k-th iterated logarithm of m is less than 1. These appear to be the densest lattices that have been explicitly constructed in high-dimensional space. New records are also established in a number of lower dimensions, beginning in dimension 96.

Let q = pn, p a rational prime, and let be the finite field with q elements. The polynomial ring is considered as an analogue of the ring of rational integers ℤ. Completing the quotient field with respect to the normalized valuation at ∞, and then taking algebraic closure, we obtained the field k∞ whose elements will be called “numbers”.

We consider the Falkner–Skan equation

in the special case β = – 1. It is known that this equation may be integrated twice to get the Riccati equation

Before turning to the questions to be considered in this paper, we recall two other problems. Let C(a, p) be the class of all convex discs of area not less than a given constant a and perimeter not greater than a given constant p. What is the densest packing and what is the most economical covering of the Euclidean plane with discs from C(a, p)?

Both problems are interesting only if p2/a < 8√3, i.e. if p is less than the perimeter of a regular hexagon of area a. In this case, the densest packing arises from a regular hexagonal tiling by rounding off the corners of the tiles by equal circular arcs so as to obtain smooth hexagons of area a and perimeter p.

If C and Co are two convex bodies in Ed we say that C slides (rolls) freely inside Co if the following condition is satisfied: for each x ∈ ∂C0 (and each rotation R) there is a translation t such that, if gC = C + t (= RC + t), then gC ⊂ Co and x ∈ ∂gC. This work establishes certain topological conditions which ensure the free rolling and sliding of C inside Co. One consequence of these conditions is that, if ∂K ∩ int gK is a topological ball for all rigid motions g, then K is a ball in the geometrical sense.

Denote by N(t) the number of zeros β + iγ of ζ(s), s = σ + it, with 0 < β < 1 and 0 < γ < t. It is well known that

where

The classical mean value theorem for Dirichlet's polynomials states that

see H. L. Montgomery [7]. This formula is very useful in the theory of the Riemann zeta-function ζ(s). From the approximate functional equation

where | χ(½ + it)| = 1, u, v ≥ 1, 2πuv = t (see E. C. Titchmarsh [8]) it follows that χ(½ + it) can be well approximated by Dirichlet's polynomials of length N< t½.